In this article we determine the global geometry of the planar
quadratic differential systems with a weak focus of third order. This
class plays a significant role in the context of Hilbert's 16-th
problem. Indeed, all examples of quadratic differential systems with
at least four limit cycles, were obtained by perturbing a system in
this family. We use the algebro-geometric concepts of divisor and
zero-cycle to encode global properties of the systems and to give
structure to this class. We give a theorem of topological
classification of such systems in terms of integer-valued affine
invariants. According to the possible values taken by them in this
family we obtain a total of $18$ topologically distinct phase
portraits. We show that inside the class of all quadratic systems
with the topology of the coefficients, there exists a neighborhood of
the family of quadratic systems with a weak focus of third order and
which may have graphics but no polycycle in the sense of \cite{DRR}
and no limit cycle, such that any quadratic system in this
neighborhood has at most four limit cycles.